Equation of State Michael Palmer Equations of State • Common ones we’ve heard of • • • • Ideal gas Barotropic Adiabatic Fully degenerate • Focus on fully degenerate EOS • Specifically on corrections made to this EOS White Dwarfs • Physics of WD’s provide a lot of opportunity to improve EOS of a fully degenerate gas • Fully degenerate core • Partially degenerate towards surface • Crystallization (starting in the interior) • Physics of crystallization Crystallization • Interior of WD, ions can not be treated as ideal • Must consider Coulomb interactions • Electrons not effect at screening ions favourable for ions to rearrange themselves • When 3kT/2 is of the order of -Ze • 3kT/2 => Thermal energy • -Ze => Coulomb energy per ion Crystallization • Coulomb coupling ratio • = (Ze)2 / (rikT) (Kippenhahn and Weigert, pg 134) – ri is the mean separation between ions » Defined as (3/(4ni))1/3 » ni is the number density of ions – k is the Boltzmann constant and T is temperature • Gives insight into the strength of the Coulomb interactions • If >> 1, kinetic energy’s role not significant and ions will try and settle into a lower energy state • Crystallization at ≈ 175 for one component plasma • Intermediate values results in phase transition from a gas to a liquid Crystallization • Tm ≈ (Z2e2/ ck)(4/30mu)1/3 critical temperature obtained from and density relation (Kippenhahn and Weigert, pg 134) • Phase from liquid to solid can not be gradual • Symmetry properties • First order phase transition => lose latent heat • Phase transition found to be first order (Winget et al. 2009) PUT IN FIGURE!!!! • Latent heat ~ kT per ion, slows cooling • Observed as bump => PUT IN FIGURE!!!! Crystallization • Lattice ions oscillate • Coulomb energy -EC = 2Z/(A1/3)61/3keV » As T-> 0 ions not at rest • • • • Oscillate about points of equilibrium Frequency E2 ~ Z2e2n0/m0 ZEzp = 3Eh(bar)/2 and Ezp = (0.6/A)61/2keV Ezp << EC so does not contribute as much to E = E0 + EC + Ezp ~ E0 + EC • Find EC influences the pressure by lowering it as compared to an ideal Fermi gas – From P -dE / d(1/n) Crystallization • Specific heat Cv effect • When << 1 the ions in the interior of the white dwarf behave as an ideal gas, Cv = 3k/2 • As ions form lattice, energy goes into lattice oscillations, results in additional degrees of freedom which raise Cv to a maximum of 3k • = 2cvMT / 5L Crystallization • Electron polarization, Coulomb crystal • fie = -f∞(xr)[1 + A(xr)(Q()/ )8] » » » » » » » » fie is the correction to free energy f∞(xr) = b1√(1 + b2/x2) A(xr) = (b3 + a3x2) / (1 + b4x2) Q() classically defined as ≈ q is TP/T xr = pF/mec b1,b2,b3,b4,a3 are all constants q = 0.205 • Form of Q() is redefined as • Q() = [ln(1 + e(qh)2)]1/2 [ln(e - (e - 2)e-(qh)2)]-1/2 PC EOS & MESA • Low temperature high density region • Carefully handles mixtures of carbon and oxygen • Accounts for all corrections to EOS laid out earlier and more • Ex => Inverse Beta Decay • Can handle both classical and quantum Coulomb crystals, Coulomb liquid interactions (weak or strong coupling),… • Default for > 80